Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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2answers
30 views

Taylor expansion of at a point different from $0$: should the variable be changed?

Find the Taylor expansion of $\arcsin x$ at point $1$. Can we change variable to get the series at point $0$? If yes how, and when do we change again to get back to $1$? More generally Let's ...
3
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0answers
22 views

One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
3
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1answer
15 views

Theorem 3.44 in Baby Rudin: Can we replace the coefficients with their absolute values?

Here's Theorem 3.44 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Suppose the radius of convergence of $\sum c_n z^n $ is $1$, and suppose $c_0 \geq c_1 \geq c_2 ...
2
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4answers
132 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
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2answers
18 views

How to find radius of convergence with power series from differential equations

So I have a question that says find the radius of convergence after I have found the power series solution of a given differential equation. I know to find the radius of convergence you take $$ ...
0
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3answers
45 views

How do I express $f(z)= \frac{6z}{z^2 - 4z + 13}$ as a power series centered at 0?

I am having trouble solving this power series problem because I usually go about decomposing the $f(z)$ and then using geometric series, but the method doesn't seem to work with this because I get ...
1
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2answers
31 views

What's wrong with my radius of convergence test?

Given $\sum \limits _{n=2} ^\infty \frac{(\ln x)^n} n$, find its radius of convergence $R$. Using the ratio test, I arrived at $$|\ln x| < 1 \implies e^{|\ln x|} < e^x \implies |x| < e ...
1
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3answers
70 views

Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...
0
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0answers
11 views

rewriting product of power series

According to $$\Lambda(\tau;q)=B_0(\tau)+\sum_{i=1}^\infty B_i (\tau) q^i$$ we define $$[\Lambda(\tau;q)-B_0(\tau)]^m=\bigg[ \sum_{i=1}^\infty B_i (\tau) q^i ...
0
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0answers
33 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
0
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2answers
118 views

Power series expansion of $x\ln(\sqrt{4+x^2}-x)$

Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$. I know that I must find power series expansion of $\ln(\sqrt{4+x^2})$ but it doesn't help. Can anyone give me a hint? many ...
0
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1answer
22 views

Reciprocal of power series with same radius

Let $f$ be a power series $f(x)=\sum a_n x^n$ with radius $R=\limsup \frac{1}{(\sqrt{|a_n|})^\frac{1}{n}}$ defined in $]-R,R[$. Let us suppose that $|f(x)|>c$ for a given $c$. Claim: Its ...
0
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1answer
23 views

Problem with the inverse expansion

Let $q=e^{2\pi i z}$ and $t=q-12q^2+66q^3-220q^4+495q^5-...$ Then why is the inverse expansion equal to $q=t+12t^2+222t^3+...$? I also do not understand the notation here: $t$ means $t(z)$ or $t(q)$? ...
1
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1answer
25 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
2
votes
1answer
82 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le ...
1
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0answers
30 views

Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}$

I am trying to find the Laurent series of the function $$f(z)=\frac{1}{z(z-1)(z-2)}$$in the rings: 1) $0<|z-1|<1$, 2) $1<|z-1|$, 3) $1<|z-2|<2 $ First I expressed $f$ as ...
2
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0answers
26 views

function with branch cuts : the radius of convergence of its Taylor series

Let $f(z)$ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
1
vote
1answer
26 views

What is the Laurent expansion of $f(z)=\frac1{z-3}$?

What is the Laurent expansion of $f(z)=\dfrac1{z-3}$? In the region, $|z-3|>0$ ? I just computed the Laurent expansion in the region $|z|>3$ by dividing the denominator by $\dfrac1z$ and ...
11
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1answer
702 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
4
votes
2answers
82 views

Proving complex trigonometric identity using power series

Prove $2$cos$^2(z) = 1+$cos$(2z)$ using power series. I know that cos$(z) = \sum (-1)^n\frac{z^{2n}}{(2n)!}$ I also know that if $a(z) = \sum a_nz^n$ and $b(z) = \sum b_nz^n$ then $a(z)b(z) ...
2
votes
2answers
187 views

Convergence of a Complex Power Series at the radius of convergence

I am currently reviewing some complex analysis, and have come across this question which I absolutely have no idea on how to attempt: Suppose the radius of convergence of the power series $f(z) = ...
3
votes
2answers
228 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of $H$’s and $T$’s. Let $N$ denote the number of tosses until you see “$TH$” for the first time. For example, for the sequence $HTTTTHHTHT$, we needed ...
0
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0answers
34 views

how to prove uniform convergence of truncated product $\cos_n(z)$ to $\cos(z)$ in the strip $\Im(z)<1$?

The function $\cos(x)$ can be expressed as an infinite product in terms of its zeros $$\cos(z)=\prod_{k=0}^{\infty}\left(1-\frac{z^2}{((2k+1)\pi/2)^2}\right)\tag{1}$$ Let us define ...
2
votes
2answers
50 views

How do you show that a power series of $e^x$ at a point converges to $e^x$?

My task is this: Find Taylor-expansion of $e^x$ at the point $1$, the convergence interval $I$ and then show that the series converges to $e^x$. My work so far: By using $Te^x @ x= 1$ we should get ...
1
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2answers
22 views

Designing a Power Series with certain $R$

Out of interest, is there a way to design a series with a certain radius of convergence? For example, $R=8$, or is there a way to turn a series for which the Radius of Convergence is known, then ...
0
votes
1answer
47 views

Writing the product $\sum\limits_{r=0}^\infty \frac{z^r}{r!} \sum\limits_{s=0}^\infty \frac{z^{-s}}{s!}$ as a power series in $z$

My lecturer states that the product $$\sum_{r=0}^\infty \frac{z^r}{r!} \sum_{s=0}^\infty \frac{z^{-s}}{s!}$$ can be written as (with $n = r-s$) $$\sum_{n=0}^\infty z^n\sum_{r=n}^\infty ...
0
votes
1answer
8 views

Radius of convergence of complex power series using Cauchy's integral formula

I have a question as follows. Let $$f(z)=\frac{\sin z}{(z-1-i)^2}$$ and $$a_n=\frac{f^{(n)}(0)}{n!}$$ Determine the radius of convergence of $$\sum_{n=0}^{\infty}a_nz^n$$ In my class we have ...
1
vote
1answer
37 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
2
votes
1answer
39 views

Radius of convergence from recurrence with variable coefficients

I am solving via power series the ivp $$y'-2xy=0,\quad y(1)=2.$$ The "solution" is $$y(x)=2\left(1+2(x-1)+3(x-1)^2+\frac{10}{3}(x-1)^3+\frac{19}{6}(x-1)^4+\frac{26}{10}(x-1)^5+\cdots\right)$$ with ...
0
votes
1answer
696 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
0
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0answers
25 views

Calculate the radius of convergence of the following power series

Let the power serie $\sum_{k\ge0}a_k(z-a)^k$ have the radius of convergence $\rho=t\in\mathbb{R^+}$, and let $p\in\mathbb{N}$. What is the radius of the following series: a) ...
1
vote
2answers
41 views

To simplify the series of matrices

Let $A$ be a square matrix in the form $A=B+O(h^2)$, where $B$ is a fixed matrix, and $O(h^2)$ is a matrix with very small elements. Assume that: $$(I-A)^{-1}=I+A+A^2+A^3+...$$ How can I esimate the ...
2
votes
1answer
53 views

Does $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ converge at the endpoints of the convergence radius?

My task is this: Find the convergence radius of$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n.$$ My work so far: By ratio test we get ...
1
vote
1answer
32 views

Frobenius Method recurrence relations

Q: By seeking a power series solution to $$2xy′′+(3−x)y′−y = 0$$ about $x=0$ show that there are two linearly independent solutions that have the recurrence relations $$a_{n+1} =\frac{a_n}{2n+3}$$ ...
0
votes
2answers
44 views

Find the fourth Taylor polynomial of f(x)=ln(x+1) at x=1

Let $f(x)=\ln(x+1)$ then (a) find the fourth Taylor polynomial of f at x=1 and (b) use part (a) find the approximate the value of ln(2.2) correct 4 decimal (c) Find an estimate for the error in ...
0
votes
2answers
33 views

Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
2
votes
1answer
37 views

Does this matrix series have an answer?

I'm trying to solve this series: $$\displaystyle\sum_{i=0}^{k}A^i B C^{k-i}$$ Where A, B, and C are $N\times N$ symmetric matrices. And $A$ and $C$ have spectral radii smaller than or equal to 1, ...
0
votes
2answers
52 views

Condition for convergence of infinite sum $\sum_{k=1}^{\infty}\frac{x^k}{k} $

Consider the following: $Q= \displaystyle\sum_{k=1}^{\infty}\frac{x^k}{k} $. What condition is required for $x$ so that $Q$ becomes convergent?
1
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2answers
39 views

Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I ...
0
votes
1answer
43 views

Interchanging limit and double series

I have a generating function $$U(z,w)=\sum_{j=0}^{\infty}\sum_{n=0}^{\infty}u_{j,n}z^jw^n$$ Where $0<z<1$, $0<w<1$, $0\leq u_{j,n}<1$. Is is true for this series that $$\lim_{z \to ...
0
votes
0answers
13 views

Hadamard's theorem; redefining indexing variable

I have seen in a few proofs the use of Hadamard's theorem to prove convergence of series like this: $\sum_{n\geq 0}z^{n!}$, or $\sum_{n\geq 0}z^{n^2}$ through simply changing the variable of indexing ...
1
vote
1answer
18 views

Annular regions in which the Laurent series converges

For the series $$\sum^\infty_{-\infty}\frac{z^n}{3^n + 1}$$ Determine the annular region in which this series converges. I understand that $\sum^\infty_{-\infty}\frac{z^n}{3^n + 1}$ can be split into ...
-2
votes
1answer
61 views

Find the value of $\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}$ [closed]

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
2
votes
3answers
35 views

Explanation of the Sum of an Infinite Series Equation

I've been presented with the following infinite sum (where $P$ is the probability of an event, and $1-P$ is therefore the probability of it not occurring. I was given the following equation as fact: ...
1
vote
1answer
27 views

Find all the $z \in \Bbb{C}$ such that the following series converges:

Find all the $z \in \Bbb{C}$ such that the following series converges: $$\sum_{n=0}^{\infty}\frac{(z+i)^{3n}}{(n^3 + 1)^{1/3}e^{3n}}$$ To solve this problem I proceed as usually, first of ...
2
votes
1answer
18 views

Complex variable: studying convergence of series in terms of radius of a different series

Trying to solve this problem: If the radius of convergence of the power series $$\sum_{n=0}^\infty a_n z^n$$ is R, with $0 < R < \infty$, then the radius of convergence $R_1$ of the ...
1
vote
1answer
27 views

Finding power series and ROC of complex function

I have the function $f(z) = \frac{3iz-6i}{z-3}$ I need to find a power series $\sum c_n (z-1)^n$ about $z_0 = 1$ I can rewrite $f$ as $\frac{2i-iz}{1-\frac{z}{3}}$, where I'm guessing the ROC ...
0
votes
0answers
20 views

Evaluation of a series with absolute value

I want to estimate or evaluate the series $$S(\xi)=\sum_{n=1}^\infty\beta_n\left|\sin(\pi n \xi)\right|,~~ \xi\in(l_0,l_1)$$ with $\beta_n=\frac{\omega\sin\left(\pi^2 n^2 ...
1
vote
1answer
65 views

What is this infinite summation?

We encountered an function defined by the infinite summation as shown below: $$F(x,a):=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{-n}(2a-n)x^{n-1}\Gamma(2a+1)}{a(2a-1)\Gamma(2a+1-n)}$$ Where ...
1
vote
3answers
60 views

Calculate $\sum_{n=0}^\infty(n+2)x^n$

I am trying to calculate $\sum_{n=0}^\infty(n+2)x^n$. I was thinking it is like the second derivative of $x^{n+2}/(n+1)$ but I am not sure how to go about calculating it. Any hints?